an object is kept at a distance of 10 cm. from a simple microscope such that magnifying power produce bymicroscope is maximum,then focal length of microscope is

1)10cm. 2)25cm. 3)16.67cm. 4)35.22cm.

To determine the focal length of a simple microscope when the object is placed at a distance of 10 cm and the magnifying power is at its maximum, we can use the formula for the magnifying power of a simple microscope. The magnifying power (M) is given by the formula:

Understanding Magnifying Power

The magnifying power of a simple microscope can be expressed as:

• M = 1 + (D / f)

Here, D is the least distance of distinct vision (typically around 25 cm for a normal human eye), and f is the focal length of the microscope.

Setting Up the Equation

Given that the object distance (u) is 10 cm, we want to find the focal length (f) that maximizes the magnifying power. To maximize M, we can rearrange the formula:

• Maximizing M occurs when the object is at the focal point, which means:
• u = f

Calculating the Focal Length

Since the object is placed at 10 cm, we can substitute this value into our equation:

• 10 cm = f

This suggests that the focal length could be 10 cm. However, we need to consider the maximum magnification condition, which occurs when the image is formed at the least distance of distinct vision (D = 25 cm).

Using the Magnifying Power Formula

Substituting D = 25 cm into the magnifying power formula:

• M = 1 + (25 / f)

To find the focal length that maximizes M when the object is at 10 cm, we can set the object distance equal to the focal length:

• 10 = f

However, we need to ensure that the image formed is at D = 25 cm. Rearranging the magnifying power equation gives:

• f = 25 / (M - 1)

Finding the Correct Option

To find the focal length that gives maximum magnification, we can substitute the known values into the magnifying power equation:

• Setting M = 3 (a reasonable assumption for maximum magnification):
• f = 25 / (3 - 1) = 25 / 2 = 12.5 cm

However, since we are looking for the closest option from the given choices, we can analyze the options:

• 1) 10 cm
• 2) 25 cm
• 3) 16.67 cm
• 4) 35.22 cm

Among these, the focal length that would allow for maximum magnification while keeping the object at 10 cm is approximately 16.67 cm, as it allows the image to be formed at a distance that is still within the range of distinct vision.

Final Answer

Thus, the correct answer is option 3) 16.67 cm. This focal length ensures that the microscope operates effectively at the specified object distance while maximizing the magnifying power.